
As \(\lbrace \gamma _k\rbrace _{k=1}^n\)  forms a martingale difference sequence, the Burkholder inequality (Lemma 2.12 of [1]}) yields
\( \phantom{\,.} \mathbf {E}|s_n(z) - \mathbf {E}s_n(z) |^2 \lesssim \frac{1}{n^2} \sum _{k=1}^n \mathbf {E}| \gamma _k |^2 \,. \) 
